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Theorem tgptmd 19609
Description: A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
Assertion
Ref Expression
tgptmd  |-  ( G  e.  TopGrp  ->  G  e. TopMnd )

Proof of Theorem tgptmd
StepHypRef Expression
1 eqid 2441 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
2 eqid 2441 . . 3  |-  ( invg `  G )  =  ( invg `  G )
31, 2istgp 19607 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e. TopMnd  /\  ( invg `  G )  e.  ( ( TopOpen `  G )  Cn  ( TopOpen
`  G ) ) ) )
43simp2bi 999 1  |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1761   ` cfv 5415  (class class class)co 6090   TopOpenctopn 14356   Grpcgrp 15406   invgcminusg 15407    Cn ccn 18787  TopMndctmd 19600   TopGrpctgp 19601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-nul 4418
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-iota 5378  df-fv 5423  df-ov 6093  df-tgp 19603
This theorem is referenced by:  tgptps  19610  tgpcn  19614  tgpsubcn  19620  tgpmulg  19623  oppgtgp  19628  tgplacthmeo  19633  subgtgp  19635  clsnsg  19639  tgpt0  19648  prdstgpd  19654  tsmssub  19682  tsmsxp  19688  trgtmd2  19702  nlmtlm  20233  qqhcn  26356
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